On Mon, 14 Apr 2008 19:07:25 -0700 (PDT), Eric Hughes wrote:
> On Apr 14, 12:52 pm, "Dmitry A. Kazakov" <mail...@[EMAIL PROTECTED]
>
> wrote:
>> I think it is wrong to consider N and universal_integer equivalent.
>
> Sure. It's {\bb Z} and universal_integer that are equivalent.
>
> Seriously, we just disagree about this. I can't take
> universal_integer seriously as a root class, because it's impossible
> to write down any representation of it.
Yes, because it is not what you wanted it be.
>> Subseting is not a sufficient condition for a
>> successful modeling.
>
> In a discussion that's got a lot of formal logic in it, the word
> "model" already means something pretty specific, usually involving a
> Tarski structure.
That is beside the point. A subset of values is not yet a type. The
requirement to be a subset (of values) alone does not enforce anything
useful.
>> Actually it is the opposite, a perfect subset cannot
>> be a good model.
>
> There was paper from the fifties (sorry, no reference handy), which
> used Turing machines to compute a Dedekind cut. On input a rational
> number, it returned one of the two symbols "<=x" or ">=x". (You
> cannot compute exact trichotomy without solving the halting problem.)
> In any language describing these machines, there's a least one (Kleene
> minimization), so there's a unique representative of such a machine
> for every computable real number, which means there's a subset
> bijection. Addition, multiplication, and their inverses are defined
> in terms of the underlying operand-machine (it's pretty easy coding,
> actually). So there's pretty close to a perfect model of the real
> numbers, whose only real limitation is that run times are horrendously
> slow. But it's also completely exact, with no compromises but
> execution speed and representation size of a machine.
>
> The subset is every real number that's computable. About as good as
> you can do with computers, I'd say.
No. It cannot represent pi and e. Now consider the following:
type A_Subset_Of_R is (pi, e); -- Ready!
The above is a perfect subset of R containing both pi and e. Isn't it
good?
>> The best models of R aren't even close to subsets. For
>> example intervals with rational bounds.
>
> In the precise meaning of model, it's just not a model, because
> there's no total ordering on such intervals, so the ordering axioms
> are not satisfied. In an imprecise meaning, it's real numbers plus
> some other concept, which is more than {\bb R}.
For any model there are properties which are not satisfied. Intervals are
good because they preserve *interesting* properties and provide fair
approximations for properties lost. Total ordering is mere a property,
which has a minor im****tance to numeric computations. In fact, each second
handbook on numeric methods starts with something like "never ever compare
reals."
--
Regards,
Dmitry A. Kazakov
http://www.dmitry-kazakov.de
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